Differential geometry of moduli spaces of quiver bundles
Abstract
Let P be a parabolic subgroup of a simple affine algebraic group G defined over C and X a compact connected K\"ahler manifold. L. \'Alvarez-C\'onsul and O. Garc\'ia-Prada associated to these a quiver Q and representations of Q into holomorphic vector bundles on X. Our aim here is to investigate the differential geometric properties of the moduli spaces representations of Q into vector bundles on X. In particular, we construct a Hermitian form on these moduli spaces. A fiber integral formula is proved for this Hermitian form; this fiber integral formula implies that the Hermitian form is K\"ahler. We compute the curvature of this K\"ahler form. Under an assumption which says that X is a complex projective manifold, this K\"ahler form is realized as the curvature of a certain determinant line bundle equipped with a Quillen metric.
Keywords
Cite
@article{arxiv.1606.01517,
title = {Differential geometry of moduli spaces of quiver bundles},
author = {Indranil Biswas and Georg Schumacher},
journal= {arXiv preprint arXiv:1606.01517},
year = {2017}
}
Comments
arXiv admin note: text overlap with arXiv:math/0112160 by other authors. 1.Introduction, 2. Quiver bundles - Notation and fundamental properties (Summary of known results), 3. Deformation theory, 4. Fiber integral formula and K\"ahler property, 5. Determinant bundle and Quillen metric, 6. Curvature of the $L^2$ metric. To appear in Journal of Geometry and Physics